For a solid, in the form of a wire or a thin rod, Young’s modulus of elasticity within elastic limit is defined as the ratio of longitudinal stress to longitudinal strain. It is given as:
It has the unit of longitudinal stress and dimensions of [MI. -1T-2] Its unit is Pascal or N/m2.
Within elastic limit the bulk modulus is defined as the ratio of longitudinal stress and volumetric strain. It is given as:
– ve indicates that the volume variation and pressure variation always negate each other.
It is defined as the ratio of the tangential stress to the shear strain.
Modulus of rigidity is given by
The ratio of change in diameter (ΔD) to the original diameter (D) is called lateral strain. The ratio of change in length (Δl) to the original length (l) is called longitudinal strain. The ratio of lateral strain to the longitudinal strain is called Poisson’s ratio.
For most of the substances, the value of σ lies between 0.2 to 0.4 When a boay is perfectly incompressible, the value of σ is maximum and equals to 0.5.
It is the property of an elastic body by virtue of which its behaviour becomes less elastic under the action of repeated alternating deforming forces.
For isotropic materials (i.e., materials having the same properties in all directions), only two of the three elastic constants are independent. For example, Young’s modulus can be expressed in terms of the bulk and shear moduli.
The ultimate tensile strength of a material is the stress required to break a wire or a rod by pulling on it. The breaking stress of the material is the maximum stress which a material can withstand. Beyond this point breakage occurs.
Work done to stretch wire
According to the formula given by
Where F is the force needed to stretch the wire of length L and area of cross-section A, is the increase in the length of the wire.
The work done by this force in stretching the wire is stored in the wire as potential energy.
Integrating both sides, we get
Which equal to the elastic potential energy U.
Now the potential energy per unit volume is
Hence, the elastic potential energy of a wire (energy density) is equal to half the product of its stress and strain.
TABLE 9.1 Young's modulus, elastic limit and tensile strengths of some materials.
Young's modulus 109N/m2
Elastic limit 107N/m2
Tensile strength 107N/m2
TABLE 9.2 Shear modulus (G) of some common materials
G(109 Nm-2 or GPa)
TABLE 9.3 Bulk modulus (B) of some common Materials
B(109 Nm-2 or GPa)
Air (at 5TP)
1.0 * 10-4
TABLE 9.4 Stress, strain and various elastic modulus
Type of Stress
State of Mater
Two equal and opposite forces perpendicular to Opposite faces (σ = F/A)
Elongation or compression parallel to force direction (ΔL/L) (longitudinal strain)
V = (FxL) /(A » ΔL)
Two equal and opposite forces parallel to opposite surfaces [forces in each case such that total force and total torque on the body vani-shes (σs * F/A)]
Pure shear, θ
C - (F x θ) /A
Forces perpendicular every where to the surface, force per unit area (pressure) same everywhere
Volume change (compression or elongation) (ΔV/V)
B = - p/(Δ V)
liquid and gas